http://elitistjerks.com/f31/t12831-prie ... mechanics/ introduces a pretty nice compilation of the facts about our shadow friends.
A summary (see full research via the aforementioned link, research was reproduced by me as well - see the post later):

This post covers shadowfiend damage
The next one covers mana return
The next after next describes shadowfiend stats

The main point about shadowfiend is that:
Shadowfiend does shadow aligned melee damage and inherits a percentage of stats from the caster

Thus,
-- Armor reduction debuffs do not affect shadowfiend's performance
-- Shadow/magic damage increasing debuffs on the target increase damage of the fiend
-- Cloak of shadows doesn't mitigate the fiend's damage
-- Shadow resistance will mitigate the fiend's damage (23 dps loss on the target with 70 SR in comparison with 0 SR)

Damage of the fiend comes from its attack power*:
-- 14 AP --> +1 dps

Shadowfiend's attack power is dynamically influenced by:
-- .57 caster shadow dmg --> +1 AP
-- 1 shadowfiend's strength --> +2 AP

So the following buffs will help the shadowfiend:
-- Blessing of Kings
-- Blessing of Might
-- Strength of Earth
-- Wrath of Air
-- Battle Shout
-- Unleashed Rage

Other ways to increase shadowfiend damage:
-- Heroism will help shadow fiend dramatically, tho only if it's cast after the pet is summoned.
-- Same goes for Drums of Battle.
-- [needs testing] Spell damage from WoA totem converts directly into the shadowfiend damage per hit (i.e. 1 spelldmg ~ .65 dps)
-- Shadowfiend does not inherit a percentage of priest's attack power
-- Being Power Infused, the fiend hits 20% faster.
-- Power Infusion doesn't affect the pet when cast on the priest

* This analysis doesnt take into account the crit% factor - its initial value, impact from agility buffs.

How all this stuff relates to the mana gained by the priest?
Someone, who has enough patience to read through my 1st post now wonders - "what on earth does this guy post? what is the practical value of all this crap?". This question is answered here.

During its lifetime fiend makes
-- 11 attacks total, with 1.5s between swings (not affected by caster's haste, but affected by bloodlust and drums)
Damage of each attack is calculated using the rules above.
Each point of damage dealt results in:
-- 2.5 times mana returned

Lets make some (rough) estimates. According to EJ:
AP = shadow_dmg*.565 + 289
DPS = AP/14 + 58.8 = shadow_dmg*0.04 + 79.4
Source: http://elitistjerks.com/506359-post43.html
Technique to validate this estimate: http://elitistjerks.com/506359-post39.html

So, suppose an ideal variant (no resistances, no dodges/parries/blocks, 15 seconds life-span):
Dmg over 15 sec = shadow_dmg*.6 + 1191
Mana returned = shadow_dmg*1.5 + 2977

So we can find out the scaling coefficients of mana returned:
-- 1.5x for shadow dmg
-- 2.7x for pet AP
-- 24.4x for WoA spd

Thus (rounding the numbers, assuming a 1.5k shadow dmg priest),
-- The full shadowfiend's lifespan (not including crits) returns more than 5k mana to the priest

Following effects benefit this:
-- Icon of the Silver Crescent gives 230 extra mana on use
-- Battle Shout gives 820 extra mana (+120 if talented, +200 with solarian trinket)
-- BoM gives: 600 extra mana
-- Heroism gives 1500 extra mana
-- WoA totem gives 2500 extra mana [needs testing]

Conclusions
-- Concerning mana regen it's almost irrelevant when to blow IotSC cooldown - automatically via a macro or wait till shadowfiend comes.
-- It's beneficial to have raid +AP effects on the pet (up to 1.5k extra mana gained = 25 mp5)
-- It's very beneficial to summon shadowfiend before a well-timed heroism (1.5k mana = 25 mp5)
-- [needs testing] It seems to be EXTREMELY beneficial to have WoA effect on the pet

* This analysis doesnt take into account the crit% factor - its initial value, impact from agility buffs.

A live test
I tested the fiend using the technique described in the EJ post: http://elitistjerks.com/506359-post39.html
Edit. More detailed report: viewtopic.php?p=133447#p133447

Strength = 153
Agility = 108 (+0.00% melee crit)
Sta = 280 + .3x my sta
Int = 133 + .3x my int (+0.00% spell crit)
AP = -20 + 2x pet str + .57x my shadow dmg
DPS = 1/14 AP
AC = 5366 + 1x pet agi + .35x my AC

Alas, neither melee nor spell crit percentages weren't shown at all. So this question is a subject to a further investigation.

Reserved.

Are you sure about this point? Most melee based elemental attacks (e.g. from elementals) ignore armor entirely.

I don't know for sure that shadowfiends follow the same rules, but it seems likely.

It seems to follow according to http://elitistjerks.com/459215-post6.html
However, I've added a "needs testing" mark near this statement.

don't forget to feed ur pet http://www.wowhead.com/?item=33874 right after it spawns. moar dmg = moar mana:)

Great idea!

lol

well a rule of thumb is sent fiend to a CoS/misery/shadweav mob boss while its not any cleaves or aoe for the next few seconds. All others are minor details that shouldnt feel our heads with.

Should we add on Improved Divinie Spirit?

So if it's affected by spell dmg how come I get more mana from it when being holy than when shadow? And it's not rarely it's always.

shadow:
1030 bonus dmg and healing
9012 mana
316/134 mp5
I usually get 3000-4500 mana

holy:
760 bonus dmg
1992 bonus healing
11012 mana
534/325 mp5
I usually get 4500-7500 mana

I hardly doubt you are getting the 7k mana you suggest in healing gear. I have 2 priests myself, only shadow with 1280+ Shadow Damage unbuffed and a Disc Priest with 1700+healing. I notice my Shadowfiend generally returns far more mana on my shadow priest than my Disc Priest.

Misery will not effect the shadowfiend's damage, but shadoweaveing and CoS will.

Are you actually counting up how much mana is returned from the combat log, or just going by how much your mana bar moves?

Shadowfiend does not get bonus damage from healing.

Shadowfiend is a shadow spell, and it is better for a shadow priest then a holy priest.